Approximation Hierarchies for Copositive Tensor Cone

TL;DR

This paper explores copositive tensors, extending the concept from matrices, and introduces hierarchies of approximations for the copositive tensor cone using various regimes like simplicial partitions and polynomial conditions.

Contribution

It provides a detailed analysis of copositive tensors, establishes conditions for their equivalence with positive semidefinite tensors, and develops multiple hierarchies for approximating the copositive tensor cone.

Findings

Hierarchies approximate the copositive tensor cone from inside and outside.

Conditions identified when copositive tensors coincide with positive semidefinite tensors.

Relationships among different approximation hierarchies are discussed.

Abstract

In this paper we discuss copositive tensors, which are a natural generalization of the copositive matrices. We present an analysis of some basic properties of copositive tensors; as well as the conditions under which class of copositive tensors and the class of positive semidefinite tensors coincides. Moreover, we have describe several hierarchies that approximates the cone of copositive tensors. The hierarchies are predominantly based on different regimes such as; simplicial partition, rational griding and polynomial conditions. The hierarchies approximates the copositive cone either from inside (inner approximation) or from outside (outer approximation). We will also discuss relationship among different hierarchies.